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Full text of "Men Of Mathematics"

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With the opening of the new century Cantor's work gradually
came to be accepted as a fundamental contribution to all
mathematics and particularly to the foundations of analysis*
But unfortunately for the theory itself the paradoxes and anti-
nomies which still infect it began to appear simultaneously.
These may in the end be the greatest contribution which
Cantor's theory is destined to make to mathematics, for their
unsuspected existence in the very rudiments of logical and
mathematical reasoning about the infinite was the direct in-
.spiration of the present critical movement in all deductive
reasoning. Out of this we hope to derive a mathematics which
is both richer and 'truer' - freer from inconsistency - than the
mathematics of the pre-Cantor era.

Cantor's most striking results were obtained in the theory of
non-denwnerable sets, the simplest example of which is the set
of all points on a line-segment. Only one of the simplest of his
conclusions can be stated here. Contrary to what intuition
would predict, two unequal line-segments contain the scans
number of points. Remembering that two sets contain the same
number of things if, and only if, the things in them can be
paired off one-to-one, we easily see the reasonableness of

Cantor's conclusion* Place the unequal segments AB9 CD as in
the figure. The line OPQ cuts CD in the point P, and AB in
Q; P and Q are thus paired off. As OPQ rotates about O, the